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A047530 Numbers that are congruent to {0, 1, 3, 7} mod 8.

%I A047530 #32 Dec 20 2021 08:00:59
%S A047530 0,1,3,7,8,9,11,15,16,17,19,23,24,25,27,31,32,33,35,39,40,41,43,47,48,
%T A047530 49,51,55,56,57,59,63,64,65,67,71,72,73,75,79,80,81,83,87,88,89,91,95,
%U A047530 96,97,99,103,104,105,107,111,112,113,115,119,120,121,123
%N A047530 Numbers that are congruent to {0, 1, 3, 7} mod 8.
%C A047530 Numbers n such that the n-th homotopy group of the topological group O(oo) does not vanish [see Baez]. Cf. A195679.
%C A047530 The a(n+1) determine the maximal number of linearly independent smooth nowhere zero vector fields on a (2n+1)-sphere, see A053381. - _Johannes W. Meijer_, Jun 07 2011
%H A047530 John C. Baez, <a href="https://doi.org/10.1090/S0273-0979-01-00934-X">The Octonions</a>, Bull. Amer. Math. Soc., Vol. 39, No. 2 (2002), pp. 145-205; <a href="https://doi.org/10.1090/S0273-0979-05-01052-9">Errata</a>, ibid., Vol. 42, No. 2 (2005), p. 213; <a href="http://math.ucr.edu/home/baez/octonions/">alternative link</a>.
%H A047530 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,1,-1).
%F A047530 From _Johannes W. Meijer_, Jun 07 2011: (Start)
%F A047530 a(n) = ceiling(n/4) + 2*ceiling((n-1)/4) + 4*ceiling((n-2)/4) + ceiling((n-3)/4).
%F A047530 a(n+1) = A053381(2^p). (End)
%F A047530 G.f.: x^2*(1+2*x+4*x^2+x^3) / ((1+x)*(x^2+1)*(x-1)^2). - _R. J. Mathar_, Oct 08 2011
%F A047530 From _Wesley Ivan Hurt_, May 21 2016: (Start)
%F A047530 a(n) = a(n-1)+a(n-4)-a(n-5) for n>5.
%F A047530 a(n) = (8n-9+i^(2n)+(2+i)*i^(-n)+(2-i)*i^n)/4, where i=sqrt(-1).
%F A047530 a(2n) = A047522(n), a(2n-1) = A047470(n). (End)
%F A047530 E.g.f.: (2 + sin(x) + 2*cos(x) + (4*x - 5)*sinh(x) + 4*(x - 1)*cosh(x))/2. - _Ilya Gutkovskiy_, May 21 2016
%F A047530 Sum_{n>=2} (-1)^n/a(n) = (8-3*sqrt(2))*log(2)/16 + 3*sqrt(2)*log(2+sqrt(2))/8 - (sqrt(2)-1)*Pi/16. - _Amiram Eldar_, Dec 20 2021
%p A047530 A047530 := proc(n): ceil(n/4) + 2*ceil((n-1)/4) + 4*ceil((n-2)/4) + ceil((n-3)/4) end: seq(A047530(n), n=0..47); # _Johannes W. Meijer_, Jun 07 2011
%p A047530 A047530:=n->(8*n-9+I^(2*n)+(2+I)*I^(-n)+(2-I)*I^n)/4: seq(A047530(n), n=1..100); # _Wesley Ivan Hurt_, May 21 2016
%t A047530 Table[(8n-9+I^(2n)+(2+I)*I^(-n)+(2-I)*I^n)/4, {n, 80}] (* _Wesley Ivan Hurt_, May 21 2016 *)
%o A047530 (PARI) a(n)=n>>2<<3+[-1,0,1,3][n%4+1] \\ _Charles R Greathouse IV_, Jun 09 2011
%Y A047530 Cf. A008621. - _Johannes W. Meijer_, Jun 07 2011
%Y A047530 Cf. A047470, A047522, A053381, A195679.
%K A047530 nonn,easy
%O A047530 1,3
%A A047530 _N. J. A. Sloane_
%E A047530 More terms from _Wesley Ivan Hurt_, May 21 2016